Partial Orders and the Matrix R in Matrix Analytic Methods

Abstract

This paper studies the matrix R, which is the minimal nonnegative solution to a nonlinear matrix equation, raised in matrix analytic methods. Based on some partial orders defined on the transition matrix of Markov chains of GI/M/1 type, the monotonicity of the corresponding matrix R and its Perron--Frobenius eigenvalue is investigated. The results are useful in estimating tail probabilities of stationary distributions of Markov chains of GI/M/1 type and constructing upper bounds for the matrix R. Applications to the GI/MAP/1 queue are discussed as well.